General Ordinary Differential Equations Research Articles

Conditions of Existence and Uniqueness of Solutions of the Multipoint Boundary Value Problem for a System of Generalized Ordinary Differential Equations

Effective sufficient conditions are established for the solvability and unique solvability of the boundary value problem \(\begin{gathered} dx(t) = dA(t) \cdot f(t,x(t)), \hfill x_i (t_i ) = \varphi _i (x){\text{ }}(i = 1,...,n), \hfill \end{gathered} \) where \(x = (x_i )_{i = 1}^n ,{\text{ }}A:[a,b] \to R^{n \times n} \) is a matrix-function with bounded variation components, \(f:[a,b] \times R^n \to R^n \) is a vector-function belonging to the Caratheodory class corresponding to \(A;t_1 ,...,t_n \in [a,b]{\text{ and }}\varphi _1 ,...,\varphi _n \) are continuous functionals (in general nonlinear) defined on the set of all vector-functions of bounded variation.

On the Correctness of Nonlinear Boundary Value Problems for Systems of Generalized Ordinary Differential Equations

Abstract The concept of a strongly isolated solution of the nonlinear boundary value problem dx(t) = dA(t) · f (t, x(t)), h(x) = 0, is introduced, where A : [a, b] → Rn×n is a matrix-function of bounded variation, f : [a, b] × Rn → Rn is a vector-function belonging to a Carathéodory class, and h is a continuous operator from the space of n-dimensional vector-functions of bounded variation into Rn . It is stated that the problems with strongly isolated solutions are correct. Sufficient conditions for the correctness of these problems are given.

Galois theory of aigebraic and differential equations

This paper will be the first part of our works on differential Galois theory which we plan to write. Our goal is to establish a Galois Theory of ordinary differential equations. The theory is infinite dimensional by nature and has a long history. The pioneer of this field is S. Lie who tried to apply the idea of Abel and Galois to differential equations. Picard [P] realized Galois Theory of linear ordinary differential equations, which is called nowadays Picard-Vessiot Theory. Picard-Vessiot Theory is finite dimensional and the Galois group is a linear algebraic group. The first attempt of Galois theory of a general ordinary differential equations which is infinite dimensional, is done by the thesis of Drach [D]. He replaced an ordinary differential equation by a linear partial differential equation satisfied by the first integrals and looked for a Galois Theory of linear partial differential equations. It is widely admitted that the work of Drach is full of imcomplete definitions and gaps in proofs. In fact in a few months after Drach had got his degree, Vessiot was aware of the defects of Drach’s thesis. Vessiot took the matter serious and devoted all his life to make the Drach theory complete. Vessiot got the grand prix of the academy of Paris in Mathematics in 1903 by a series of articles.

Parallel Implementations of Iterated Runge-Kutta Methods

We investigate different parallel algorithms for the iter ated Runge-Kutta method on distributed memory mul tiprocessors for the solution of systems of ordinary differential equations (ODEs). The iterated Runge-Kutta method is an iteration scheme for the numerical solu tion of initial-value problems of nonstiff ODEs; embed ded approximation formulae are used to control the step size. The parallel algorithms realize potential par allelism across the method and the system in a group- SPMD (single-program, multiple-data) programming style, using an appropriate set of communication prim itives that can be implemented on all common topolo gies. A theoretical performance analysis with run-time formulae and a run-time simulation show the value of the algorithms. The implementation on the Intel iPSC/ 860 confirms the predicted run times. The speedup values depend strongly on the particular system of ODEs to be solved. The parallel iterated Runge-Kutta method is applied to a typical discretization problem, the discretized Brusselator equation. Application-spe cific modifications of the general parallel ODE solver are developed, which result in a considerable reduc tion in the parallel execution time.

High-Order Numerical Method for Two-Point Boundary Value Problems

In this paper the two-point boundary value problem is transformed into general first-order ordinary differential equation system through introduction of conditions of an integral character to supplement the simultaneous set of first-order equations. A new discrete approximation of a high-order compact difference scheme is presented for the first-order system. It is a block-bidiagonal profile and removes the limits of other high-order discrete schemes at the interval ends. The numerical tests of a seventh-order compact difference scheme show that the proposed scheme is very convenient and efficient for linear and nonlinear two-point boundary value problems.

Pseudo Almost Periodic Solutions of Some Differential Equations, II

We first present some results about pseudo almost periodic functions. Then we use these results to show some theorems about existence and uniqueness of the pseudo almost periodic solutions of the linear and the quasi-linear nonautonomous equations. Finally we set up some theorems of the existence of pseudo almost periodic solutions implying the existence of almost periodic solutions of general ordinary differential equation systems.

On the Correctness of Linear Boundary Value Problems for Systems of Generalized Ordinary Differential Equations

Abstract The sufficient conditions are established for the correctness of the linear boundary value problem dx(t) = dA(t) · x(t) + df(t); l(x) = c 0, where and are matrix- and vector-functions of bounded variation, , and l is a linear continuous operator from the space of n-dimentional vector-functions of bounded variation into .

Integrable types of nonlinear ordinary differential equations of higher-orders

In this papers some integrable types of more general nonlinear ordinary differential equations of higher-orders are proposed in virtue of Leibnitz formula, and formulas of higher-order derivatives of the composite functions as well as substitution variables. The expressions for the general integrations of some of the equations are presented. The results obtained are the generalization of those in the references. Finally, some examples are also given.

NEW METHOD FOR SOLVING PARTIAL AND ORDINARY DIFFERENTIAL EQUATIONS USING FINITE-ELEMENT TECHNIQUE

In this paper I introduce a new method for solving partial and ordinary differential equations with large first, second, and third derivatives of the solution in some part of the domain using the finite-element technique (here called the Galerkin-Gokhman method) The method is based on the application of the Galerkin method to a modified differential equation. The exact solution of the modified equation is the Galerkin approximation for the unknown function with exact values of the unknown at the nodal points. An application of the Galerkin-Gokhman method to a general second-order nonlinear ordinary differential equation and to Navier-Stokes equations in the case of developing flow in a pipe is formulated. I also include the results of an application of the Galerkin-Gokhman method to two specific ordinary differential equations. One is y — dy/dx = 0, the other one is a second-order nonlinear equation describing fully developed turbulent flow in a pipe.

Degenerate bifurcations and catastrophe sets via frequency analysis

We present some bifurcation conditions using the well-known stability analysis of feedback systems. A general ordinary differential equation system is formulated in two parts: one that considers the linear part and the other that includes the memoryless nonlinear part, in a similar way as the describing function. The bifurcation conditions are obtained using the results of the generalized Nyquist stability criterion (GNSC) with some explicit formulae derived from some properties of the complex variable We analyse simultaneously both static and dynamic (Hopf) bifurcations and their degeneracies in a rich example, a continuous stirred-tank reactor (CSTR), in which two consecutive, irreversible, first-order reactions A→B→C occur

Solution of general ordinary differential equations using the algebraic theory approach

AbstractThe algebraic theory for numerical methods, as developed by Herrera [3–7], provides a broad theoretical framework for the development and analysis of numerical approximations. To this point, the technique has only been applied to ordinary differential equations with constant coefficients. The present work extends the theory by developing a methodology for equations with variable coefficients. Approximation of the coefficients by piecewise polynomials forms the foundation of the approach. Analysis of the method provides firm error estimates. Furthermore, the analysis points to particular procedures that produce optimal accuracy. Example calculations illustrate the computational procedure and verify the theoretical convergence rates.

A comparison theorem for general nonlinear ordinary differential equations

A comparison theorem for general nonlinear ordinary differential equations

A Flux-Limited Diffusion Model for Charged-Particle Transport

AbstractA new algorithm for modeling charged-particle transport in a fully ionized plasma is presented. A standard multigroup discretization of the Fokker-Planck-Boltzmann equation is transport-corrected to implicitly include the anisotropic effects of both coulomb scattering and nuclear reactions. This allows the subsequent application of the Levermore flux-limited diffusion theory, which was originally developed for isotropic radiative transfer calculations. A finite differencing of the resulting spatial transport operator is constructed so as to yield centered and upwinded operators in the diffusion and free-streaming limits, respectively. The time integration is performed by the general purpose ordinary differential equation solver TORANAGA. This approach results in a highly vectorizable algorithm that has been implemented on the CRAY-1. Some numerical results are presented that compare this algorithm to the corresponding, but far more expensive, Monte Carlo calculations.

Cancer chemotherapy: optimal control using the Verhulst-Pearl equation.

General (deterministic) ordinary differential equations for the representation of cancer growth are presented when the growth is perturbed due to the action of a chemotherapeutic agent. The Verhulst-Pearl equation is introduced as a particular example of a growth equation applicable to human tumors. An optimal control problem with general performance criterion and state equation is formulated and shown to possess a novel feedback control relationship. This relationship is used in two continuous drug delivery problems involving the Verhulst-Pearl equation.

Cancer chemotherapy: Optimal control using the Verhulst-Pearl equation

General (deterministic) ordinary differential equations for the representation of cancer growth are presented when the growth is perturbed due to the action of a chemotherapeutic agent. The Verhulst-Pearl equation is introduced as a particular example of a growth equation applicable to human tumors. An optimal control problem with general performance criterion and state equation is formulated and shown to possess a novel feedback control relationship. This relationship is used in two continuous drug delivery problems involving the Verhulst-Pearl equation.

Theory of nonhomogeneous linear ordinary differential equations

One constructs a general formula for the solution of the Cauchy problem in the case of a general linear ordinary differential equation of an arbitrary order.

Invariance properties in Hermite-Padé approximation theory

The Padé approximant is invariant under both linear fractional transformations of the function value and linear fractional transformations (origin fixed) of the argument. The paper reports on an investigation of these invariance properties to the class of Hermite-Padé approximants based on the class of general ordinary nonlinear differential equations. Further, the paper characterizes the class with these invariance properties. Uniqueness is discussed. Examples are given, such as the Padé-Riccati approximants, and allowed types of singular points are discussed.

A Second Order Superlinear Oscillation Criterion

AbstractA new oscillation criterion is given for general superlinear ordinary differential equations of second order of the form x″(t)+ a(t)f[x(t)]=0, where a ∈ C([t0∞,)), f∈C(R) with yf(y)>0 for y≠0 and and f is continously differentiable on R-{0} with f'(y)≥0 for all y≠O. In the special case of the differential equation (γ > 1) this criterion leads to an oscillation result due to Wong [9].

A Connection Problem for a Regular and an Irregular Singular Point of Complex Ordinary Differential Equations

In recent papers [SIAM J. Math. Anal., 11 (1980), pp. 848–862, 863–875] D. Schmidt and the author studied the connection problem for two neighboring regular singular points for quite general linear complex ordinary differential equations. In the present paper these results are generalized in the case that one singular point is irregular singular, but of rank 1 and the leading matrix has n distinct eigenvalues; one singular point is regular singular; and there may be several other singular points. The main result is a limit formula for those connection coefficients which are essential in some specified way. An application to the generalized Heun equation is given.

Generalized multitime expansions for equations with slowly varying coefficients

The successive terms in a uniformly valid multitime expansion of the solutions of constant coefficient differential equations containing a small parameter ϵ may be obtained without resorting to secularity conditions if the time scales ti = ϵit(i = 0, 1, …) are used. Similar results have been achieved in some cases for equations with variable coefficients by using nonlinear time scales generated from the equations themselves. This paper extends the latter approach to the general second order ordinary differential equation with slowly varying coefficients and examines the restrictions imposed by the method.